deletions | additions       diff --git a/Magnetic Trapping.tex b/Magnetic Trapping.tex  index 3b88584..52d3f99 100644  --- a/Magnetic Trapping.tex  +++ b/Magnetic Trapping.tex 
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{\bf v}_g & \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat{n}}_\perp + \frac{\omega^2}{N k_\perp} {\bf \hat{r}} \bigg] \nonumber \\  & \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat {n}}_\perp + v_A {\bf \hat{r}} \bigg] \, .  \end{align}  Below $r_{\rm MG}$, the group velocity of Alfven waves is $v_A {\bf \hat{B}}$. \hat{B}}$, in the direction of the magnetic field. Thus, although the radial group velocity of the incoming magneto-gravity waves is comparable to that of Alven waves, their horizontal group velocity is much larger than the Alfven velocity. Coupling to Alfven waves therefore requires a large change in both direction and magnitude of the group velocity. This may cause most of the gravity wave energy to reflect at $r_{\rm MG}$ rather than being transmitted into Alfven waves.  Let's first consider waves that are transmitted at $r_{\rm MG}$. The incoming magneto-gravity waves couple to pure Alfven modes in the cavity $r